Mukhi S., Mukunda N. Lectures on Advanced Mathematical Methods for Physicists
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32
Chapter
2.
Homotopy
N
N
N
Figure 2.7: a
is
deformable
to
a-Ion
S2
Ie.
Now let us show
that
this
loop a
is
homotopic
to
its
inverse
a-I
(see
Fig. 2.7).
Simply keep
the
end
points fixed
and
deform
the
path
so
that
it goes
down
the
other
side of
the
sphere.
By
the
identification map,
this
is
the
same
as
a
path
going upwards from S
to
N along
the
front of
the
sphere, namely,
a-I.
Thus
we have shown
that
in
this
case,
(2.15)
So
[a]
*
[a]
=
[a
2
]
=
[a]
*
[a]-1
=
[i]
(2.16)
It
is
clear
that
a shrinkable loop based
at
N
(which
is
closed
already
in
S2)
and
the
loop
a(t)
we have
just
described (which
is
closed only on
S2
Ie)
represent
all
the
homotopy classes
of
S2
Ie.
Thus
(2.17)
Under
the
homomorphism:
[i]
->
0,
[a]->
1
(2.18)
this
becomes
the
additive group
Z2
of integers modulo
2.
This
is
an
example of
a finite
homotopy
group:
7rl
(S2
I
e)
=
Z2.
(v)
T2
=
SI
X
SI.
This
is
the
2-torus. As visualized in
our
own 3-dimensional
world,
this
looks like Fig. 2.8.
To find
its
fundamental group, we use
the
theorem
stated
earlier,
about
7rl
of a
product
space.
Thus
we have
7rl(T2)
=
Z ffi
Z.
Similarly, for
the
n-torus
Tn
=
SI
@
SI
@
...
@
SI
(n
times),
we
have
7rl(Tn)
= Z ffi
Zffi
...
ffi Z
(n
times).
For
the
two-torus, a homotopy class is labelled by a
pair
of integers
(nl,
n2)
which described
the
number
of times a loop winds
around
the
first
and
second
SI
respectively.
Exercise:
Describe some loops
on
the
torus
which lie in
the
homotopy classes
(1,0), (0,1), (1,1).
Exercise:
Find
a space
S
with
7rl
(S)
=
Z ffi
Z2
(recall
the
theorem
on
7rl
of a
2.3.
Homotopy Type and Contractibility
.
33
Figure 2.8:
The
two-torus
8
1
X
8
1
.
product
space).
(vi)
m?
-
B2:
this
is
the
plane
with
a disc
cut
out
of it. We
can
find
its
11"1
directly,
but
instead
let us show
that
this space is
of
the
same homotopy
type
as
8
1
.
This
is a nice example
of
two spaces which
are
not
homeomorphic
but
are
of
the
same
homotopy
type
.
Figure
2.9:
m?
minus a disc is
of
the
same
homotopy
type
as
8
1
.
This
is
illustrated
in Fi
g.
2.9. Define
f :
m?
-
B2
---+
8
1
to
be
the
map
which sends every
point
x
Em?
-
B2
to
the
corresponding
unit
vector
X,
which
lies
on
8
1
.
In
other
words, simply
project
the
point
down along
the
line joining
it
to
the
origin until
it
reaches
the
unit
circle, which defines its image in
8
1
.
The
reverse
map
9 :
8
1
---+
lR,
2
-
B2
takes all points of
the
circle
8
1
and
puts
them
on
the
boundary
of
the
disc removed from
lR,
2
.
Clearly
this
is a homotopy
of
topological spaces. We
can
image "continuously shrinking"
lR,
2
-
B2
down
to
the
circle.
Thus
11"1
(lR,2
-
B2)
=
11"1
(8
1
)
=
Z.
(vii)
lR,2
minus two discs (see Fig.2.10).
We will show
that
11"1
of
this
space is
non-abelian.
Consider
the
two loops
a,
{3
based
at
Xo
as in
the
first figure
of
Fig. 2.10. Clearly
they
are
not
homotopic
to
each
other,
so
[a]
=f.
[
{3
].
Now consider
the
products
[a]
*
[
{3
]
= [a
*
{3
]
=
'Y
and
[
{3
]
*
[a]
=
[
{3
*
a]
=
6
in
the
same figure.
It
is evident
that
'Y
=f.
5~
and
therefore
[a]
*
[
{3
]
=f.
[
,8]
*
[a
].
Thus
11"1
(8)
is a non-Abelian group in this case.
This
group is known as
the
free
group
on two generators, which simply
means
its
elements
are
'
abstract
products
of two independent generators ar-
ranged
in
any
order,
with
different orderings representing independent elements
34
Chapter
2.
Homotopy
Figure 2.10: Two based loops
(3,
a;
a
loop,
in
the
class
[a
*
.8
];
a loop
[)
in.
the
class
[
(3
*
a]
(in
m?
with
two discs removed).
of
the
group.
To see
that
7rl
can
be
non-Abelian in general, consider multiplication of
classes
of
based loops
[
a],
[
(3
]
in
the
two possible orders as depicted in Fig.
2.1l.
Keeping
the
base
point
fixed, one
cannot
in general deform
the
second
diagram
a.
'"
a.
*
~:
~
[
I
0
112
~
~
*
a.:
a.
'"
I
1
0
112
Figure 2.11:
The
product
of loops
a,
(3
in different orders. Clearly
the
two
products
are
not
deformable into each
other
.
of Fig. 2.11 into
the
first.
This
is
the
basic reason why
7rl
(8)
is non-Abelian in
general.
2.4
Higher
H;omotopyGroups
We have seen
that
the
fundamental group
7rl
(8)
is
the
collection
of
homotopy
classes
of
continuous
maps
from
the
closed interval [0, 1]
to
the
space
8,
with
the
requirement
that
the
map
be
"based"
at
a given point
Xo.
This
requirement
is implemented
by
the
condition
that
the
end-points
of
the
closed interval
be
mapped
into a common
point
in
8.
In
this section, we generalize
the
notion
of
loops
to
higher-dimensional objects ;mapped into a topological space.
This
will
allow us
to
define more homotopy groups associated
to
a given space.
Each
one
2.4.
Higher Homotopy Groups
35
could
then
potentially
capture
some new topological information, which would
lead us closer
to
the
goal of
und
erstand
ing
the
topology of a space
through
the
study
of homotopy.
A simple generalization of a based loop
is
obtained
by considering maps
from
the
closed
unit
square
[0
,
1]
@
[0
,
1]
to
S,
with
the
requirement
that
the
entire
boundary
of
the
square
be sent
to
a common
point
in
X.
Thus
consider
a:
[0,1]
@
[0,1]
---->
S
(2.19)
given by a continuous function
a(tl,
t2)
where
tl
is
in
the
first [0,1]
and
t2
is
in
the
second
[0
, 1]' such
that
(2.20)
I
I
al
l
l
·
1
2)
~
12
•
Xo
,---
!
II
0
I
I
/
Figure 2.12: A based "two loop" maps
the
boundary
of
[0
,
1]
x [0,1]
to
a single
point
Xo.
Such a
map
may
be
considered
the
two-dimensional
ana
logue of
the
based
loops described earlier. L
et
us call
it
a
2-1oop.
Now a
homotopy
between two
2-loops
a(tl,
t2)
and
{3(t
l
,
t2)
based
at
the
same
point
Xo
is
defined by:
H:
[0
,
1]
@
[0
,
1]
@
[0
,
1]
---->
S
H(s
,
t
l
,
t2)
is
a continuous
map
with
the
following properties:
H(O,
t
l
,
t2)
=
a(tl,
t2)
H(l,
tl
,
t2)
=
{3(h,
t2)
H(s,
t
l
,
t
2
)\
=
Xo
for a
lls.
(t,
or
t2
=
Oor
1)
(2.21 )
(2.22)
This
map
describes a continuous deformation of one 2-loop into
another,
anal-
ogous
to
what
was done for loops in
standard
homotopy.
All this
is
easily generalized
to
"n
-loops" ,
though
one should keep in mind
that
"n-loop"
is
not
standard
mathematical
terminology.
Definition:
An
n-loop
in X based
at
Xo
is
a continuous map:
a:
[0
,
l
]n
---->
X
36
Chapter
2.
Homotopy
such
that
a(tl,
...
,t
n
)!
=XO
(any
ti
=
0
or
I)
Here,
the
nth
power of
the
interval
[0,
1]
denotes
its
Cartesian
product
with
itself n times.
Definition:
A
homotopy
of
n-loopB
a(iI,
...
,
tn)
and
(J(tl,
...
,
tn)
is
a continu-
ous
map
H:
[O,I]n+l
~
S
such
that
H(s;tl,
...
,tn)!
=a(tl,
...
,t
n
)
s=o
H(s,
tl"'"
tn)!S=1
=
(J(h,
...
,
tn)
H(s,
tl,
...
,
tn)!
=
Xo
for all s
any
ti
=
0
or
I
This
homotopy
is
an
equivalence relation, as one
can
easily check.
With
the
above definitions, we follow
the
predictable
path
of defining
the
product
of
generalised n-Ioops, leading finally
to
the
concept of
the
nth
homotopy group
7r
n
of a space.
Definition:
The
product
of two n-Ioops
a,
(J
is
given by
0<
t
<
~
- - 2
1
- <
t
<
1
2 - -
In
a previous section
on
loops, we
had
constructed
the
inverse of a loop
by
tracing
the
loop backwards. Now
that
we
are
dealing
with
"n-Ioops",
an
analogous
operation
can
be defined
by
simply
tracing
backwards in
the
first
argument
tl
of
the
loop, leaving
the
others
unchanged.
Definition:
The
inverse
of
an
n-loop
a
is
given by
a-l(tl,""
tn)
=
a(I
- iI,
t2,
...
, tn)
With
all these properties it becomes evident
that
equivalence classes of
based n-Ioops form a group,
just
as for simple loops.
Definition:
The
nth
homotopy
group
of
the
space
S,
with
base
point
Xo,
denoted
7rn(S,
xo),
is
7rn(S,
xo)
= {
[a]
I
a
is
an
n-Ioop based
at
Xo
}
As
before,
[a]
denotes
the
equivalence class of all loops homotopic
to
a,
and
the
group
operations
are
just
as for I-loops.
In
particular,
the
identity
2.4.
Higher
Homotopy
Groups
37
element of
the
group corresponds
to
the
homotopy class
of
the
constant
loop,
a(tl'
."
..
,
tn)
=
Xo
for all
ti.
For path-connected spaces,
7r
n
(8,
xo)
is independent
of
Xo
as one would expect. "
Example:
7r2(8)
is
the
group of homotopy classes
of
2-100ps. A 2-100p is
just
the
image
of
the
2-sphere 8
2
in
the
given topological space.
It
is easy
to
see
that
in lR
3
,
all 2-spheres
can
be
deformed
to
a point, so
7r2
(lR
3
)
=
{i}.
On
the
other
hand
in
:ni
3
-
{O},
a 2-sphere enclosing
the
origin
cannot
be
shrunk. So
7r2
(lR
3
-
{O})
is non-trivial.
In
fact (although
it
is
not
as easy
to
see intuitively
as for
the
analogous case
7rl
(lR
2
-
{O})),
it
turns
out
that
(2.23)
Before giving more examples, let us
state
an
important
theorem.
Theorem:
The
homotopy groups
7r
n
(8),
n
~
2,
are
Abelian.
Proof:
Instead
of giving a formal proof let us use diagrams
to
indicate how
this
works.
The
relevant
diagram
for 7r2 is Fig. 2.13. Let us see how
a
*
f3
a
I
::=0
(3
a*(3:
[[J
I
[[J
Figure 2.13:
The
product
of two-loops
a,
f3
in different orders.
can
be
homotopically deformed
to
f3
*
a
in
this
case. Note
that
a
and
f3
are
constrained
to
map
all
the
boundary
points
to
Xo.
Thus
the
product
a*
f3
maps
the
boundary
of
the
rectangle
to
Xo.
The
trick now
is
to
continuously modify
the
parametrisation
of
the
map
a*
f3
while leaving
its
image
in
the
topological space
unchanged. For a I-loop we could only speed
up
or
slow down
the
"traversal
speed"
in
terms
of
the
parameter
t,
of each
part
of
the
loop."
But
here,
with
two
parameters,
we
can
do something more drastic.
We
start
by
homotopically deforming
a*
f3
to
map
many
more
points in
the
rectangle
to
xo,
namely, all those within some distance of
the
boundary. This
is
illustrated
in
Fig
. 2.14, where
the
whole
shaded
region is
mapped
to
Xo.
This
38
Chapter
2.
Homotopy
III
Figure 2.14:
The
map
a*f3
deformed so
that
the
entire
shaded
region is
mapped
to
the
same point
X
o.
operation
is continuous as long as
the
regions marked
1
and
2,
over which
the
map
is
non-constant, are non-empty regions.
Next, perform
the
sequence of operations shown in Fig. 2.15. Clearly all
the
2
a
a
2
Figure
2.15:
Proof
that
7r2
is Abelian.
operations are continuous
and
during
the
whole process,
the
image
of
the
map
remains
the
same.
But
at
the
end
we have
managed
to
change
the
parametri-
sation
of
the map
from
that
of
a
*
f3
to
that
of
f3
*
a.
Thus
the
two 2-loops
are
homotopic
and
belong in
the
same equivalence class. Hence,
(2.24)
and
we have shown, as desired,
that
7r2(S)
is
an
Abelian group.
Exercise:
Convince yourself
that
the
same procedure does
not
work for
7rl,
and
that
it
works for all
n
2:
2.
To conclude this section, we
state
without
proof
a few useful results which
tend
to
occur in problems of physical interest. ·
(i)
7r
n ( sn)
=
Z
(ii)
7rm(sn)
=
0, m
<
n
2.4.
Higher
Homotopy
Groups
39
(iii)
7r3(S2)
=
Z
The
interested reader should consult
the
mathematical
literature
for proofs of
these
and
many
more results
on
homotopy groups.
All homotopy groups
7rn(S)
of a topological space are invariant
under
home-
omorphisms,
and
hence should
be
thought
of as topological invariants charac-
terising
the
space.
They
cannot
be
changed except by discontinuous (non-
homeomorphic) deformations of
the
space.
This page intentionally left blankThis page intentionally left blank
Chapter
3
Differentiable
Manifolds
I
Starting
with
a topological space,
it
is sometimes possible
to
put
additional
structure
on
it
which makes
it
locally look like
Euclidean
space
IR
n
of
some
dimension
n.
On
this
space, by imposing
suitable
requirements,
it
will
then
be
possible
to
differentiate functions.
The
space so
obtained
is called a differentiable
manifold.
This
notion
is
of
central
importance
in
General Relativity, where
it
provides a
mathematical
description
of
spacetime.
3.1
The
Definition
of
a
Manifold
Among
the
topological spaces we have studied,
the
Euclidean
space
IR
n
=
IR
Q9
IR
Q9
.•.
Q9
IR
is special. Besides being a
metric
space
and
hence a topological
space
with
the
metric
topology,
it
has
an
intuitive
notion
of
"dimension" , which
is simply
the
number
n
of
copies
of
IR.
Let us
concentrate
on
two simple cases:
IR,
the
(one-dimensional) real line,
and
IR
2
,
the
(two-dimensional) plane.
If
the
dimension
of
a space is
to
be
topologically meaningful, two different
IR
n
,
IR
m
should
not
be
homeomorphic
to
each
other,
for m
=I-
n.
This
is indeed
the
case.
Let
us
demonstrate
that
IR
is
not
homeomorphic
to
IR2.
Assume
the
con-
trary:
suppose
there
is a homeomorphism (recall
that
this
means a 1-1 contin-
uous function
with
continuous inverse)
f :
IR
-+
IR
2
.
Consider
the
restriction
of
this
function:
f :
IR
-
{O}
-+
IR2
-
{O}
where we have deleted
the
point
{O}
from
IR,
and
its image
under
f,
which we define
to
be
the
origin, from
IR2.
Now if
f :
IR
-+
IR2
is a
homeomorphism
then
so is
f :
IR-
{O}
-+
IR2
-
{O}.
But
IR
-
{O}
is disconnected, while
IR2
-
{O}
is clearly still connected. Since
connectivity is invariant
under
homeomorphism,
this
is a contradiction.
Thus
the
original
homeomorphism
f :
IR
-+
IR2
does
not
exist,
and
IR
is
not
home-
omorphic
to
IR2.
Similarly, one finds
that
IR
n
and
IR
m
are
not
homeomorphic
for m
=I-
n.
So "dimension" is indeed a topological property.